Page 446 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 427

Note that (t, z) are the coordinates of an optical impulse with the origin (0, 0) at the transmitter. If a receiver
is placed at Z = z, T denotes the time at the receiver if the receiver clock is shifted from the transmitter clock
by the time of flight z. Since Z and T are functions of z and t,wehave
1
T T
= 1, =− , (10.61)
1
t z
Z Z
= 0, = 1, (10.62)
t z
q q Z q T
= +
z Z z T z
q q
= ⋅ 1 + (− ), (10.63)
1
Z T
q q Z q T
= +
t Z t T t
q
= ⋅ 1, (10.64)
T
2
q ( q ) Z ( q ) T
= +
t 2 Z t t T T t
2
q
= . (10.65)
T 2
Substituting Eqs. (10.63)–(10.65) in Eq. (10.58), we find
2
q(T, Z) q(T, Z) − q(T, Z) q(T, Z)
2
+ (− )= q(T, Z)− 1 − i
1
Z T 2 T 2 T 2
or
2
q q iq
2
i − =− . (10.66)
Z 2 T 2 2
Eq. (10.66) describes the propagation of the optical field envelope in a fiber when the nonlinear effects are
ignored. Eq. (10.66) is equivalent to Eq. (10.56). For a Gaussian input, the output electric field envelope is
given by Eq. (2.158),
√ ( )
P T T 2
0 0
q(T, Z)= exp − , (10.67)
T 1 2T 2
1
2
T =(T − i Z) 1∕2 . (10.68)
1 0 2
Let
T = |T | exp (i ), (10.69)
1 1 1
where
4
2 2 1∕2
2
|T | =(T + Z ) , (10.70)
1
2
0
( )
1 −1 Z
2
=− tan . (10.71)
1 2
2 T
0